Average Error: 13.2 → 0.2
Time: 37.6s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)}\]
\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4241673 = lambda1;
        double r4241674 = lambda2;
        double r4241675 = r4241673 - r4241674;
        double r4241676 = sin(r4241675);
        double r4241677 = phi2;
        double r4241678 = cos(r4241677);
        double r4241679 = r4241676 * r4241678;
        double r4241680 = phi1;
        double r4241681 = cos(r4241680);
        double r4241682 = sin(r4241677);
        double r4241683 = r4241681 * r4241682;
        double r4241684 = sin(r4241680);
        double r4241685 = r4241684 * r4241678;
        double r4241686 = cos(r4241675);
        double r4241687 = r4241685 * r4241686;
        double r4241688 = r4241683 - r4241687;
        double r4241689 = atan2(r4241679, r4241688);
        return r4241689;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4241690 = phi2;
        double r4241691 = cos(r4241690);
        double r4241692 = lambda2;
        double r4241693 = cos(r4241692);
        double r4241694 = lambda1;
        double r4241695 = sin(r4241694);
        double r4241696 = r4241693 * r4241695;
        double r4241697 = cos(r4241694);
        double r4241698 = sin(r4241692);
        double r4241699 = r4241697 * r4241698;
        double r4241700 = log1p(r4241699);
        double r4241701 = expm1(r4241700);
        double r4241702 = r4241696 - r4241701;
        double r4241703 = r4241691 * r4241702;
        double r4241704 = phi1;
        double r4241705 = cos(r4241704);
        double r4241706 = sin(r4241690);
        double r4241707 = r4241705 * r4241706;
        double r4241708 = sin(r4241704);
        double r4241709 = r4241691 * r4241708;
        double r4241710 = r4241693 * r4241697;
        double r4241711 = r4241709 * r4241710;
        double r4241712 = r4241698 * r4241695;
        double r4241713 = r4241712 * r4241709;
        double r4241714 = r4241711 + r4241713;
        double r4241715 = r4241707 - r4241714;
        double r4241716 = atan2(r4241703, r4241715);
        return r4241716;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.2

    \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
  2. Using strategy rm

  3. Applied sin-diff6.8

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
  4. Using strategy rm

  5. Applied cos-diff0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\]

  6. Applied distribute-rgt-in0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)}}\]
  7. Using strategy rm

  8. Applied expm1-log1p-u0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)}\]

  9. Final simplification0.2

    \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))